   Chapter 11.9, Problem 21E

Chapter
Section
Textbook Problem

Find a power series representation for f, and graph f and several partial sums s n ( x ) on the same screen. What happens as n increases? f ( x ) = x 2 x 2 + 1

To determine

To find:

i) The power series representation for the function fx=x21+x2

ii) The graph of f  with several partial sums on same screen. Also write what happens as n  increases.

Explanation

1) Concept:

i) The power series expansion of the function

11-x=n=0xn=1+x+x2+x3+     x<1

2) Given:

fx=x21+x2

3) Calculation:

The first step is to express the function x21+x2,  as sum of a power series.

x21+x2= 11+x2·x2

We know that

11-x= n=0xn

Therefore, we can rewrite the function

11+x2=11--x2

This gives,

11-(-x2)=n=0-x2n=n=0-1nx2n

x21-(-x2)=x2n=0-1nx2n=n=0-1nx2n+2

So, the expansion of power series

n=0-1nx2n+2=x2-x4+x6-x8+x10-

This series convergent when -x2<1 i

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